Theme Essential Question: How do we use a system of equations to model real world situations that have multi-constraints to simulate relationships?
Essential Questions: How are systems of equations represented and interpreted graphically?
Standards Analyze and solve pairs of simultaneous linear equations.
8. EE.8a Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously.
8. EE.8c Solve real-world and mathematical problems leading to two linear equations in two variables. For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair.
Objectives
The student will use a graphical representation to solve and interpret the solution to a system of linear equations.
The students will investigate the outcome of linear equations that intersect, coincide, or are parallel.
(Taken from Ohio Dept of Education Mathematics Model Curriculum 6-28-2011 ) This cluster builds on the informal understanding of slope from graphing unit rates in Grade 6 and graphing proportional relationships in Grade 7 with a stronger, more formal understanding of slope. It extends solving equations to understanding solving systems of equations, or a set of two or more linear equations that contain one or both of the same two variables. Once again the focus is on a solution to the system. Most student experiences should be with numerical and graphical representations of solutions. Beginning work should involve systems of equations with solutions that are ordered pairs of integers, making it easier to locate the point of intersection, simplify the computation and hone in on finding a solution. More complex systems can be investigated and solve by using graphing technology. Contextual situations relevant to 8th graders will add meaning to the solution to a system of equations. Students should explore many problems for which they must write and graph pairs of equations leading to the generalization that finding one point of intersection is the single solution to the system of equations. Provide opportunities for students to connect the solutions to an equation of a line, or solution to a system of equations, by graphing, using a table and writing an equation. Students should receive opportunities to compare equations and systems of equations, investigate using graphing calculators or graphing utilities, explain differences verbally and in writing, and use models such as equation balances. EXAMPLE: Problems such as, “Determine the number of movies downloaded in a month that would make the costs for two sites the same, when Site A charges $6 per month and $1.25 for each movie and Site B charges $2 for each movie and no monthly fee.” Students write the equations letting y = the total charge and x = the number of movies. Site A: y = 1.25x + 6 Site B: y = 2x Students graph the solutions for each of the equations by finding ordered pairs that are solutions and representing them in a t-chart. Discussion should encompass the realization that the intersection is an ordered pair that satisfies both equations. And finally students should relate the solution to the context of the problem, commenting on the practicality of their solution. Problems should be structured so that students also experience equations that represent parallel lines and equations that are equivalent. This will help them to begin to understand the relationships between different pairs of equations: When the slope of the two lines is the same, the equations are either different equations representing the same line (thus resulting in many solutions), or the equations are different equations representing two not intersecting, parallel, lines that do not have common solutions. System-solving in Grade 8 should include estimating solutions graphically, solving using substitution, and solving using elimination. Students again should gain experience by developing conceptual skills using models that develop into abstract skills of formal solving of equations. Provide opportunities for students to change forms of equations (from a given form to slope-intercept form) in order to compare equations.
Assessment Product http://mwstrange.com/Systems%20of%20Linear%20Equations%20Group%20Project.pdf In this project the students will be choosing between two real life situations and then using systems of linear equations to decide what to buy. Please be advised computer access will be advantageous. Week 1: Teacher introduction to the project and rubric. Week 2: Students thoroughly exploring and selecting their project. Week 3: Students will conduct research and apply a graphical solution to their project (manually or using technology). Week 4: Students will support their solution by using the substitution method and/or the elimination method to their project. Week 5: Students will analyze their solution, complete their project, and conduct presentations.
Key Questions
What is the process to graphically represent a solution to a system of linear equations?
What conclusion can be made when the system of linear equation intersect, coincide, or are parallel?
3. Observable Student Behaviors
The student can graph and interpret a solution to a system of linear equations.
Mathematical Practices 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning.
Vocabulary System of equations, coordinate plane, slope, y intercept, slope-intercept form
Suggested Activities [see Legend to highlight MCO and HYS]
Houghton Mifflin OnCore Mathematics Middle School Grade 8
Unit 3.3, p. 75-78
ABC Mastering the Common Core in Mathematics 6.1- p. 76-78
Gizmo Lessons
8.EE.8a
Solving Linear Systems by Graphing
Compare a system of equations in standard form or in slope‑intercept form to its graph. Examine the graph and table of values. Determine the solutions to the system.
Special Types of Solutions to Linear Systems
Compare a system of equations in standard form to its graph. Examine the graph and table of values. Determine the solutions to the system.
Teaching the Common Core Math Standards with Hands-On Activities by Judith Muschla
Content: Solving Systems Graphically (Manually)
Dates:4/8-4/12
Theme Essential Question:
How do we use a system of equations to model real world situations that have multi-constraints to simulate relationships?
Essential Questions:
How are systems of equations represented and interpreted graphically?
Standards
Analyze and solve pairs of simultaneous linear equations.
Objectives
Background Information
Recommended: For a quick overview of the standard(s) to be addressed in this lesson, see Arizona’s Content Standards Reference Materials. http://www.azed.gov/wp-content/uploads/PDF/MathGr8.pdf
(Taken from Ohio Dept of Education Mathematics Model Curriculum 6-28-2011 )
This cluster builds on the informal understanding of slope from graphing unit rates in Grade 6 and graphing proportional relationships in Grade 7 with a stronger, more formal understanding of slope.
It extends solving equations to understanding solving systems of equations, or a set of two or more linear equations that contain one or both of the same two variables. Once again the focus is on a solution to the system. Most student experiences should be with numerical and graphical representations of solutions. Beginning work should involve systems of equations with solutions that are ordered pairs of integers, making it easier to locate the point of intersection, simplify the computation and hone in on finding a solution. More complex systems can be investigated and solve by using graphing technology.
Contextual situations relevant to 8th graders will add meaning to the solution to a system of equations. Students should explore many problems for which they must write and graph pairs of equations leading to the generalization that finding one point of intersection is the single solution to the system of equations. Provide opportunities for students to connect the solutions to an equation of a line, or solution to a system of equations, by graphing, using a table and writing an equation. Students should receive opportunities to compare equations and systems of equations, investigate using graphing calculators or graphing utilities, explain differences verbally and in writing, and use models such as equation balances.
EXAMPLE: Problems such as, “Determine the number of movies downloaded in a month that would make the costs for two sites the same, when Site A charges $6 per month and $1.25 for each movie and Site B charges $2 for each movie and no monthly fee.”
Students write the equations letting y = the total charge and x = the number of movies.
Site A: y = 1.25x + 6
Site B: y = 2x
Students graph the solutions for each of the equations by finding ordered pairs that are solutions and representing them in a t-chart. Discussion should encompass the realization that the intersection is an ordered pair that satisfies both equations. And finally students should relate the solution to the context of the problem, commenting on the practicality of their solution.
Problems should be structured so that students also experience equations that represent parallel lines and equations that are equivalent. This will help them to begin to understand the relationships between different pairs of equations: When the slope of the two lines is the same, the equations are either different equations representing the same line (thus resulting in many solutions), or the equations are different equations representing two not intersecting, parallel, lines that do not have common solutions.
System-solving in Grade 8 should include estimating solutions graphically, solving using substitution, and solving using elimination. Students again should gain experience by developing conceptual skills using models that develop into abstract skills of formal solving of equations. Provide opportunities for students to change forms of equations (from a given form to slope-intercept form) in order to compare equations.
Assessment
Product
http://mwstrange.com/Systems%20of%20Linear%20Equations%20Group%20Project.pdf
In this project the students will be choosing between two real life situations and then using systems of linear equations to decide what to buy. Please be advised computer access will be advantageous.
Week 1: Teacher introduction to the project and rubric.
Week 2: Students thoroughly exploring and selecting their project.
Week 3: Students will conduct research and apply a graphical solution to their project (manually or using technology).
Week 4: Students will support their solution by using the substitution method and/or the elimination method to their project.
Week 5: Students will analyze their solution, complete their project, and conduct presentations.
Key Questions
3. Observable Student Behaviors
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
Vocabulary
System of equations, coordinate plane, slope, y intercept, slope-intercept form
Suggested Activities [see Legend to highlight MCO and HYS]
- Houghton Mifflin OnCore Mathematics Middle School Grade 8
Unit 3.3, p. 75-78- ABC Mastering the Common Core in Mathematics 6.1- p. 76-78
Gizmo Lessons- 8.EE.8a
- Solving Linear Systems by Graphing
- Compare a system of equations in standard form or in slope‑intercept form to its graph. Examine the graph and table of values. Determine the solutions to the system.
- Special Types of Solutions to Linear Systems
- Compare a system of equations in standard form to its graph. Examine the graph and table of values. Determine the solutions to the system.
Teaching the Common Core Math Standards with Hands-On Activities by Judith Muschla- 8.EE.8 Activity 1 p. 180, Activity 2 p. 181
Glencoe Pre-Algebra – p. 414-418, p. 428Glencoe Algebra 1- p. 369-375
- Highly Recommended:
http://illustrativemathematics.org/illustrations/469 (G.8)http://illustrativemathematics.org/illustrations/472 (G.8)
http://illustrativemathematics.org/illustrations/554 (G.8)
http://illustrativemathematics.org/illustrations/73 (G.8)The Illustrative Mathematics Project offers guidance to states, assessment consortia, testing companies, and curriculum developers by illustrating the range and types of mathematical work that students will experience in a faithful implementation of the Common Core State Standards. The website features a clickable version of the Common Core in mathematics and the first round of "illustrations" of specific standards with associated classroom tasks and solutions.
Diverse Learners
Homework
Suggested:
Terminology for Teachers
Ethnicity/Culture | Immigration/Migration | Intercultural Competence | Socialization | Racism/Discrimination
High Yield Strategies
Similarities/Differences | Summarizing/Notetaking | Reinforcing/Recognition | Homework/Practice |
Non-Linguistic representation | Cooperative Learning | Objectives/Feedback |
Generating-Testing Hypothesis | Cues, Questions, Organizers
Resources
Professional Texts
Literary Texts
Informational Texts
- See New York Common Core Aligned Task (other resources)
http://schools.nyc.gov/Academics/CommonCoreLibrary/SeeStudentWork/default.htmArt, Music, and Media
Manipulatives
Games
Videos
SMART Board Lessons, Promethean Lessons
- Smartboard Resource Website Smartboard lesson search engine
- 8. EE.8a Solving Systems of Equations through Graphing
Solve systems of two linear equations in two variables by graphing. Interpret solutions of two linear equations.- 8. EE.8c Solving Problems Involving Algebraic Techniques (Question Set)
Solving Problems Involving Algebraic Techniques (Question set)Websites
Other Activities, etc.
Language
Arts
Week 1
Week 2
Week 3
Week 4
Week 5
Week 6
Matrix
Week 1
Week 2
Week 3
Week 4
Week 5
Week 6
Home K-2
Home 3-6
Home 6-8
Unit 1
Unit 2
Unit 3
Unit 4
Unit 5
Unit 6